Question

Integrate the rational functions.
$\int \frac{2x}{x^2+3x+2}dx.$

Answer 1

Let $\frac{2x}{x^2+3x+2}=\frac{2x}{x^2+2x+x+2}$
$=\frac{2x}{x(x+2)+1(x+2)}=\frac{2x}{(x+2)(x+1)}$
Let $\frac{2x}{(x+2)(x+1)}=\frac{A}{(x+2)}+\frac{B}{(x+1)}$
$\Rightarrow \frac{2x}{(x+2)(x+1)}=\frac{A(x+1)+B(x+2)}{(x+2)(x+1)}\Rightarrow 2x=Ax+A+Bx+2B\to 2x=x(A+B)+(A+2B).$
On comparing the coefficients of x and constant term on both sides, we get
A+B=2 .....(i)
and A+2B=0 .....(ii)
On subtracting Eq. (ii)form Eq. (i), we get -B =2 $\Rightarrow$ B =-2
On putting the value of B in Eq. (i),we get A-2=2 $\Rightarrow$ A =4
$\therefore \int \frac{2x}{(x+2)(x+1)}dx=\int \frac{A}{(x+2)}dx+\int \frac{B}{(x+1)}dx=\int \frac{4}{(x+2)}dx+\int \frac{(-2)}{(x+1)}dx=4log|x+2|-2log|x+1|+C$